Biserial algebras and generic bricks

Recall that $M$ in $\operatorname*{Mod}\Lambda$ is a brick if $\operatorname*{End}_{\Lambda}(M)$ is a division algebra. Then, $\Lambda$ is called brick-finite if it admits only finitely many bricks (up to isomorphism). Each brick is evidently indecomposable and by $\operatorname*{Brick}(\Lambda)$ and $\operatorname*{brick}(\Lambda)$ we denote the subsets of all bricks, respectively in $\operatorname*{Ind}(\Lambda)$ and $\operatorname*{ind}(\Lambda)$. Although each representation-finite (rep-finite, for short) algebra is brick-finite, the converse is not true in general (e.g. any representation-infinite local algebra admits a unique brick). More precisely, the notion of brick-finiteness is of interest only if $\Lambda$ is a rep-infinite tame algebra, or else when $\Lambda$ is wild but not strictly wild (a standard argument yields that any strictly wild algebra is brick-infinite).

Bricks and their properties play pivotal roles in different areas and can be studied from various perspectives (see Section 2). As we do here, the notion of brick-finiteness can be viewed as a conceptual counterpart of representation-finiteness. To better highlight this perspective, let us present two analogous characterizations that also motivate our work. Thanks to some classical and recent results, $\Lambda$ is known to be rep-finite if and only if $\operatorname*{Ind}(\Lambda)=\operatorname*{ind}(\Lambda)$, whereas it is brick-finite if and only if $\operatorname*{Brick}(\Lambda)=\operatorname*{brick}(\Lambda)$ (for details, see [Se]). Furthermore, through the lens of approximation theory, $\Lambda$ is shown to be rep-finite if and only if every full subcategory of $\operatorname*{mod}\Lambda$ is functorially finite, whereas it is brick-finite if and only if every torsion class in $\operatorname*{mod}\Lambda$ is functorially finite (for details, see [DIJ]).

Before we recall a powerful tool in the study of rep-infinite algebras, observe that each $\Lambda$-module $M$ can be viewed as a (right) module over $\operatorname*{End}_{\Lambda}(M)$. Then, the endolength of $M$ is the length of $M$ when considered as an $\operatorname*{End}_{\Lambda}(M)$-module. In particular, a $\Lambda$-module $G$ in $\operatorname*{Ind}(\Lambda)\setminus\operatorname*{ind}(\Lambda)$ is called generic if it is of finite endolength. Generic modules are known to play a significant role in representation theory of algebras. For instance, any generic $\Lambda$-module of endolength $d$ gives rise to an infinite family of (non-isomorphic) modules of length $d$ in $\operatorname*{ind}(\Lambda)$ (see [CB1] and the references therein). In fact, Crawley-Boevey [CB1] has given an elegant realization of the Tame/Wild dichotomy theorem of Drozd [Dr] in terms of generic modules and their endolength (see Theorem 4.1). Based on his characterization, one can further refine tame algebras and say $\Lambda$ is $m$-domestic if it admits exactly $m$ generic modules (up to isomorphism). In general, $\Lambda$ is domestic if $\Lambda$ is $m$-domestic for some $m\in\mathbb{Z}_{\geq 0}$.

For an algebra $\Lambda$, it is known that $\Lambda$ is rep-infinite if and only if it admits a generic module ([CB1]). Hence, it is natural to ask for an analogous characterization of brick-infinite algebras. First, observe that if $\Lambda$ is brick-finite, no generic $\Lambda$-module is a brick (see Theorem 2.1). To describe our proposal for a new treatment of brick-infinite case, we introduce some new terminology. In particular, we say $G$ is a generic brick of $\Lambda$ if $G$ is a generic module and it belongs to $\operatorname*{Brick}(\Lambda)$. Furthermore, $\Lambda$ is called brick-continuous if for a positive integer $d$, there exists an infinite family in $\operatorname*{brick}(\Lambda)$ consisting of bricks of length $d$. The latter notion is motivated by the algebro-geometric properties of bricks (see Section 4). In particular, $\Lambda$ is said to be brick-discrete if it is not brick-continuous. We remark that brick-discrete algebras are studied in [CKW], where the authors call them “Schur representation-finite” algebras and view them as a generalization of rep-finite algebras. In [Mo2], it is conjectured that brick-discrete algebras are the same as brick-finite algebras.

In the rest of this section, we focus on our main problem and present our results. In particular, our conjecture below is primarily inspired by our earlier results in [Mo1, Mo2] on the behavior of bricks, as well as our work on minimal $\tau$-tilting infinite algebras in [MP] (see also Conjecture 2.1).

To verify the above conjecture in full generality, one only needs to treat those algebras which are brick-infinite and minimal with respect to this property. In particular, we say $\Lambda$ is a minimal brick-infinite algebra (min-brick-infinite, for short) if $\Lambda$ is brick-infinite but all proper quotient algebras of $\Lambda$ are brick-finite. Thus, a concrete classification of such algebras will be helpful in the study of our conjecture. As discussed in Subsection 2.4, min-brick-infinite algebras are novel counterparts of minimal representation-infinite algebras (min-rep-infinite, for short). This classical family is extensively studied due to their role in several fundamental problems, particularly in the celebrated Brauer-Thrall Conjectures (see [Bo1] and references therein). Note that min-brick-infinite algebras also enjoy some important properties that could be helpful in the study of Conjecture 1.1 (see Theorem 2.4).

It is known that biserial algebras form an important family of tame algebras, among which string, gentle and special biserial algebras have appeared in many areas of research (for definitions and details, see Subsection 2.2). In particular, if $\Lambda$ is a special biserial algebra, one can combinatorially describe all indecomposable modules in $\operatorname*{mod}\Lambda$, as well as their Auslander-Reiten translate and morphisms between them (for details, see [BR] and [WW]). However, for arbitrary biserial algebras, there is no full classification of indecomposable modules and their representation theory is more complicated than that of special biserial algebras.

In 2013, Ringel [Ri1] gave an explicit classification of those min-rep-infinite algebras which are special biserial. Thanks to the more recent results of Bongartz [Bo1], one can show that Ringel’s classification is in fact the full list of min-rep-infinite biserial algebras. In particular, any such biserial algebra is a string algebra which is either $1$-domestic or else non-domestic (for more details, see [Ri1] and Subsection 2.4). Here, we obtain an analogue of Ringel’s classification and give a full list of min-brick-infinite biserial algebras in terms of their quivers and relations.

Due to some technical observations that are further explained in Section 4, in this paper, unless stated otherwise, we restrict to tame algebras and verify the above conjecture for all biserial algebras. Before stating our main classification result, let us fix some new terminology. In particular, for an arbitrary algebra $\Lambda$, we say $\Lambda$ is $m$-generic-brick-domestic if it admits exactly $m$ (isomorphism classes) of generic bricks. More generally, $\Lambda$ is generic-brick-domestic if it admits only finitely many generic bricks. For the definition and concrete description of generalized barbell algebras, we refer to Subsection 2.4 and Figure 2. In particular, the following theorem follows from our results in Section 3.

If $Q$ is an affine Dynkin quiver, the algebra $kQ$ is known to be min-brick-infinite and tame. A classical result of Ringel [Ri2] shows that every such path algebra admits a unique generic brick. Hence, from this point of view, the preceding theorem extends the aforementioned result of Ringel and treats some non-hereditary tame min-brick-infinite algebras (see Theorem 4.4).

We remark that the generalized barbell algebras are never domestic. However, by the above theorem, they are always $1$-generic-brick-domestic. As an interesting consequence of our classification, the following corollary is shown in Section 4. It is known that any string algebra which is not domestic must be of non-polynomial growth. Therefore, such algebras can be seen as $\infty$-domestic.

To prove the above corollary, in Section 4.2 we present several examples and as the result give an explicit algorithm to construct an $n$-generic-brick-domestic algebra for each $n\in\mathbb{Z}_{\geq 0}$. Moreover, in Example 4.10, we give a gentle algebra which is not generic-brick-domestic. The above corollary further highlights the fundamental differences between generic modules and generic bricks, as well as the domestic and generic-brick-domestic algebras (for example, see Questions 4.2).

As an important consequence of our classification result, we obtain a useful characterization of brick-finiteness of biserial algebras. Note that the rank of the Grothendieck group of $\Lambda=kQ/I$ is the number of vertices in $Q$, denoted by $|Q_{0}|$.

The above theorem asserts a stronger version of Conjecture 1.1 for the family of biserial algebras. Moreover, the numerical condition given in part $(2)$ is similar to that of Bongartz’s for the length of $1$-parameter families of indecomposable modules over rep-inf algebras (for details, see [Bo2]). This opens some new directions in the study of distribution of bricks (for example, see Question 4.1).

As mentioned earlier, any systematic study of bricks and their properties provide new insights into several other domains of research. We end this section by the following corollary which highlight these connections and postpone further applications of our results to our future work. All the undefined terminology and notations used in the next assertion appear in Sections 2 or 3. Moreover, proof of the following corollary follows from Theorem 2.2 and Corollary 4.5.

By a quiver we always mean a finite directed graph, formally given by a quadruple $Q=(Q_{0},Q_{1},s,e)$, with the vertex set $Q_{0}$ and arrow set $Q_{1}$, and the functions $s,e:Q_{1}\rightarrow Q_{0}$ respectively send each arrow $\alpha$ to its start. For $\alpha$ and $\beta$ in $Q_{1}$, by $\beta\alpha$ we denote the path of length two which starts at $s(\alpha)$ and ends at $e(\beta)$. Let $Q_{1}^{-1}:=\{\gamma^{-1}\,|\,\gamma\in Q_{1}\}$ be the set of formal inverses of arrows of $Q$. That is, $s(\gamma^{-1})=e(\gamma)$ and $e(\gamma^{-1})=s(\gamma)$.

Following our assumptions in Section 1, every algebra $\Lambda$ is an admissible quotient $kQ/I$ of a path algebra $kQ$ for some quiver $Q$, up to Morita equivalence. In this case, the pair $(Q,I)$ is called a bound quiver. All quotients of path algebras will be assumed to be admissible quotients. Moreover, modules over $\Lambda$ can be seen as representations over the bound quiver $(Q,I)$. Provided we begin from a bound quiver, this dictionary is still available and $k$ can be an arbitrary field. In this case, $\Lambda$-modules are representations of the corresponding bound quiver. For $M$ in $\operatorname*{mod}\Lambda$, let $|M|$ denote the number of non-isomorphic indecomposable modules that appear in the Krull-Schmidt decomposition of $M$. In particular, for $\Lambda=kQ/I$, we have $|\Lambda|=|Q_{0}|$, which is the same as the rank of $K_{0}(\Lambda)$, where $K_{0}(\Lambda)$ denotes the Grothendieck group of $\operatorname*{mod}\Lambda$. In particular, this rank is the number of (isomorphism classes of) simple modules in $\operatorname*{mod}\Lambda$.

For $\Lambda=kQ/I$, unless specified otherwise, we consider a minimal set of uniform relations that generate the admissible idea $I$. That is, each generator of $I$ is a linear combination of the form $R=\sum_{i=1}^{t}\lambda_{i}p_{i}$, where $t\in\mathbb{Z}_{>0}$ and $\lambda_{i}\in k\setminus\{0\}$, and all $p_{i}$ are paths of length strictly larger than one in $Q$ starting at the same vertex $x$ and ending at the same vertex $y$. For the most part, we work with monomial and binomial relations, which are respectively when $t=1$ and $t=2$. In particular, the monomial relations of length $2$, known as quadratic monomial relations, play a crucial role in the study of (special) biserial algebras. A vertex $v$ in $Q$ is a node if it is neither a sink nor a source, and for any arrow $\alpha$ incoming to $v$ and each arrow $\beta$ outgoing from $v$, we have $\beta\alpha\in I$.

In this paper, all subcategories are assumed to be full and closed under isomorphism classes, direct sum and summands. Moreover, for a given collection $\mathfrak{X}$, we say a property holds for almost all elements in $\mathfrak{X}$ if it is true for all but at most finitely many elements of $\mathfrak{X}$.

An algebra $\Lambda$ is said to be biserial if for each left and right indecomposable projective $\Lambda$-module $P$, we have $\operatorname*{rad}(P)=X+Y$, where $X$ and $Y$ are uniserial modules and $X\cap Y$ is either zero or a simple module. Biserial algebras were formally introduced by Fuller [Fu], as a generalization of uniserial algebras, and Crawley-Boevey [CB2] showed that they are always tame.

Special biserial algebras form a well-known subfamily of biserial algebras and thanks to their rich combinatorics, their representation theory is well-studied. We recall that an algebra $\Lambda$ is special biserial if it is Morita equivalent to an algebra $kQ/I$ such that the bound quiver $(Q,I)$ satisfies the following conditions:

1. (B1)

   At every vertex $x$ in $Q_{0}$, there are at most two incoming and at most two outgoing arrows.

2. (B2)

   For each arrow $\alpha$ in $Q_{1}$, there is at most one arrow $\beta$ such that $\beta\alpha\notin I$ and at most one arrow $\gamma$ such that $\alpha\gamma\notin I$.

A special biserial algebra $\Lambda=kQ/I$ with $(Q,I)$ as above is called a string algebra if $I$ in $kQ$ can be generated by monomial relations. Over string algebras, all indecomposable modules and morphisms between them are understood (see [BR] and [WW]).

An important subfamily of string algebras consists of gentle algebras. Recall that $\Lambda=kQ/I$ is gentle if it is a string algebra and $I$ can be generated by a set of quadratic monomial relations such that $(Q,I)$ satisfies the following condition:

1. (G)

   For each arrow $\alpha\in Q_{1}$, there is at most one arrow $\beta$ and at most one arrow $\gamma$ such that $0\neq\alpha\beta\in I$ and $0\neq\gamma\alpha\in I$.

Observe that if $\Lambda=kQ/I$ is biserial (respectively a string algebra, or gentle algebra), then for every $x\in Q_{0}$ and each $\gamma\in Q_{1}$ the quotient algebras $\Lambda/\langle e_{x}\rangle$ and $\Lambda/\langle\gamma\rangle$ are again biserial (respectively string, or gentle). Moreover, an arbitrary quotient of a (special) biserial algebra is again (special) biserial. For $\Lambda=kQ/I$, a string in $\Lambda$ is a word $w=\gamma_{k}^{\epsilon_{k}}\cdots\gamma_{1}^{\epsilon_{1}}$ with letters in $Q_{1}$ and $\epsilon_{i}\in\{\pm 1\}$, for all $1\leq i\leq k$, such that

1. (S1)

   $s(\gamma_{i+1}^{\epsilon_{i+1}})=e(\gamma_{i}^{\epsilon_{i}})$ and $\gamma_{i+1}^{\epsilon_{i+1}}\neq\gamma_{i}^{-\epsilon_{i}}$, for all $1\leq i\leq k-1$;

2. (S2)

   Neither $w$, nor $w^{-1}:=\gamma_{1}^{-\epsilon_{1}}\cdots\gamma_{k}^{-\epsilon_{k}}$, contain a subpath in $I$.

A string $v$ in $(Q,I)$ is serial if either $v$ or $v^{-1}$ is a direct path in $Q$. Namely, $v=\gamma_{k}\cdots\gamma_{2}\gamma_{1}$ or $v^{-1}=\gamma_{k}\cdots\gamma_{2}\gamma_{1}$, for some arrows $\gamma_{i}$ in $Q_{1}$. For a string $w=\gamma_{k}^{\epsilon_{k}}\cdots\gamma_{1}^{\epsilon_{1}}$, we say it starts at $s(w)=s(\gamma_{1}^{\epsilon_{1}})$, ends at $e(w)=e(\gamma_{k}^{\epsilon_{k}})$, and is of length $l(w):=k$. Moreover, a zero-length string, denoted by $e_{x}$, is associated to every $x\in Q_{0}$. Suppose $\operatorname{Str}(\Lambda)$ is the set of all equivalence classes of strings in $\Lambda$, where for each string $w$ in $\Lambda$ the equivalence class consists of $w$ and $w^{-1}$ (i.e. set $w\sim w^{-1}$). A string $w$ is called a band if $l(w)>0$ and $w^{m}$ is a string for each $m\in\mathbb{Z}_{\geq 1}$, but $w$ itself is not a power of a string of strictly smaller length, where each band is considered up to all cyclic permutations of it. For a vertex $x$ of $\Lambda$, we say $w$ in $\operatorname{Str}(\Lambda)$ visits $x$ if it is supported by $x$. Moreover, $w$ passes through $x$ provided that there exists a nontrivial factorization of $w$ at $x$. That is, there exist $w_{1},w_{2}\in\operatorname{Str}(\Lambda)$ with $s(w_{2})=x=e(w_{1})$, such that $l(w_{1}),l(w_{2})>0$ and $w=w_{2}w_{1}$. For $\alpha\in Q_{1}$, we say $w$ is supported by $\alpha$ if the string $w$ contains $\alpha$ or $\alpha^{-1}$ as a letter.

Let $G_{Q}$ denote the underlying graph of $Q$. Then, every string $w=\gamma_{d}^{\epsilon_{d}}\cdots\gamma_{1}^{\epsilon_{1}}$ induces a walk in $G_{Q}$, where a vertex or an edge may occur multiple times. The representation $M(w):=((V_{x})_{x\in Q_{0}},(\varphi_{\alpha})_{\alpha\in Q_{1}})$ of $(Q,I)$ associated to $w$ has an explicit construction as follows: Put a copy of $k$ at each vertex $x_{i}$ of the walk induced by $w$. This step gives the vector spaces $\{V_{x}\}_{x\in Q_{0}}$, where $V_{x}\simeq k^{n_{x}}$ and $n_{x}$ is the number of times $w$ visits $x$. To specify the linear maps of the representation $M(w)$ between the two copies of $k$ associated to $s(\gamma_{i}^{\epsilon_{i}})$ and $e(\gamma_{i}^{\epsilon_{i}})$, put the identity in the direction of $\gamma_{i}$. Namely, this identity map is from the basis vector of $s(\gamma_{i})$ to that of $e(\gamma_{i})$ if $\epsilon_{i}=1$, and it goes from the basis vector of $e(\gamma_{i})$ to that of $s(\gamma_{i})$, if $\epsilon_{i}=-1$. If $\Lambda=kQ/I$, the string module associated to $w$ is an indecomposable $\Lambda$-module given in terms of the representation $M(w)$. Note that for every string $w$, there is an isomorphism of modules (representations) $M(w)\simeq M(w^{-1})$.

To every band $v\in\operatorname{Str}(\Lambda)$, in addition to the string module $M(v)$, there exists another type of indecomposable $\Lambda$-modules associated to $v$ which are called band modules. For the description of band modules, as well as the morphisms between the string and band modules over string algebras, we refer to [BR], [Kr] and [WW].

Let $M$ be a finitely generated $\Lambda$-module. We say that $M$ is basic if no indecomposable module appears more than once in the Krull-Schmidt decomposition of $M$. We denote by $|M|$ the number of non-isomorphic indecomposable direct summands of $M$. Also, $M$ is rigid if $\operatorname{Ext}^{1}_{\Lambda}(M,M)=0$. Let $\operatorname*{rigid}(\Lambda)$ denote the set of isomorphism classes of all basic rigid modules in $\operatorname*{mod}\Lambda$. Moreover, by $\operatorname*{\mathtt{i}rigid}(\Lambda)$ we denote the set of indecomposable modules in $\operatorname*{rigid}(\Lambda)$. Similarly, $M$ is said to be $\tau$-rigid if $\operatorname{Hom}_{\Lambda}(M,\tau_{\Lambda}M)=0$, where $\tau_{\Lambda}$ denotes the Auslander-Reiten translation in $\operatorname*{mod}\Lambda$. Provided there is no confusion, we simply use $\tau$ to denote the Auslander-Reiten translation. By $\tau\operatorname*{-rigid}(\Lambda)$ and $\mathtt{i}\tau\operatorname*{-rigid}(\Lambda)$ we respectively denote the set of isomorphism classes of basic $\tau$-rigid modules and the indecomposable $\tau$-rigid modules. A rigid module $X$ is called tilting if $\operatorname*{pd}_{\Lambda}(X)\leq 1$ and $|X|=|\Lambda|$, where $\operatorname*{pd}_{\Lambda}(X)$ denotes the projective dimension of $X$. Analogously, a $\tau$-rigid module $M$ is $\tau$-tilting if $|M|=|\Lambda|$. More generally, $M$ is called support $\tau$-tilting if $M$ is $\tau$-tilting over $A/\langle e\rangle$, where $e$ is an idempotent in $A$. By $\operatorname*{tilt}(\Lambda)$ and $\tau\operatorname*{-tilt}(\Lambda)$ we respectively denote the set of all isomorphism classes of basic tilting and $\tau$-tilting modules in $\operatorname*{mod}\Lambda$. Moreover, $s\tau\operatorname*{-tilt}(\Lambda)$ denotes the set of isomorphism classes of all basic support $\tau$-tilting modules in $\operatorname*{mod}\Lambda$.

$\tau$-tilting theory, introduced by Adachi, Iyama and Reiten [AIR], has been a modern setup in representation theory of associative algebras where many rich ideas from cluster algebras and classical tilting theory meet. Through this new setting, the authors address the deficiency of classical tilting theory with respect to the mutation of tilting modules. In [AIR], the notion of mutation of clusters is conceptualized in terms of mutation of (support) $\tau$-tilting modules.

Given an algebra $\Lambda$, it is a priori a hard problem to decide whether or not the set of (support) $\tau$-tilting modules is finite. Since these modules form the main ingredient of $\tau$-tilting theory, finding explicit necessary and sufficient conditions such that an algebra has $|\tau\operatorname*{-tilt}(\Lambda)|<\infty$ is monumental. This has spurred a lot of research in this direction, among which the elegant “brick-$\tau$-rigid correspondence” appearing in [DIJ] has proved to be very useful. Some important characterizations of $\tau$-tilting finite algebras are recalled in the rest of this subsection.

Recall that a $\Lambda$-module $Y$ is called a brick if $\operatorname*{End}_{\Lambda}(Y)$ is a division algebra. That is, any nonzero endomorphism of $Y$ is invertible. As in Section 1, by $\operatorname*{Brick}(\Lambda)$ and $\operatorname*{brick}(\Lambda)$ we respectively denote the set of isomorphism classes of bricks in $\operatorname*{Mod}\Lambda$ and $\operatorname*{mod}\Lambda$. If the field $k$ is algebraically closed, then $Y$ belongs to $\operatorname*{brick}(\Lambda)$ if and only if $\operatorname*{End}_{\Lambda}(Y)\simeq k$. Such modules are sometimes called Schur representations, particularly when they are studied from the algebro-geometric viewpoint, such as in [CKW]. An algebra $\Lambda$ is called brick-finite provided $|\operatorname*{Brick}(\Lambda)|<\infty$. Meanwhile, we warn the reader that those algebras called “Schur representation-finite” in [CKW] are not known to be necessarily brick-finite (for further details on this difference, see Subsection 2.5, as well as [Mo2, Subsection 1.3]).

One of our main goals in this paper is to establish a relationship between certain modules in $\operatorname*{Brick}(\Lambda)\setminus\operatorname*{brick}(\Lambda)$ and those in $\operatorname*{brick}(\Lambda)$. In this regard, the following result of Sentieri [Se] is of interest.

We now list some of the fundamental results on $\tau$-tilting finiteness of algebras. Recall that a subcategory $\mathcal{T}$ of $\operatorname*{mod}\Lambda$ is a torsion class if it is closed under quotients and extensions. Let $\operatorname*{tors}(\Lambda)$ denote the set of all torsion classes in $\operatorname*{mod}\Lambda$. For $M$ in $\operatorname*{mod}\Lambda$, let $\operatorname{Fac}(M)$ denote the subcategory of $\operatorname*{mod}\Lambda$ consisting of all those modules that are quotients of some finite direct sum of copies of $M$. It is known that $\mathcal{T}$ in $\operatorname*{tors}(\Lambda)$ is functorially finite provided $\mathcal{T}=\operatorname{Fac}(M)$, for some $M$ in $\operatorname*{mod}\Lambda$. By $\operatorname*{f-tors}(\Lambda)$ we denote the subset of $\operatorname*{tors}(\Lambda)$ consisting of functorially finite torsion classes. The following important result relates the finiteness of the notions introduced so far. In particular, it states that an algebra is brick-finite if and only if it is $\tau$-tilting finite.

Here we collect some of our main results from [Mo1, Mo2], as well as [MP], which are used in this paper. We begin with a useful observation that is freely used in our reductive arguments. In particular, we recall that each epimorphism of algebras $\psi:\Lambda_{1}\rightarrow\Lambda_{2}$ induces an exact functorial full embedding $\widetilde{\psi}:\operatorname*{mod}\Lambda_{2}\rightarrow\operatorname*{mod}\Lambda_{1}$. Particularly, we get $\operatorname*{ind}(\Lambda_{2})\subseteq\operatorname*{ind}(\Lambda_{1})$ and also $\operatorname*{brick}(\Lambda_{2})\subseteq\operatorname*{brick}(\Lambda_{1})$. This implies that if $\Lambda_{2}$ is rep-infinite (respectively, brick-infinite) then so is $\Lambda_{1}$. Thus, by Theorem 2.2, $\tau$-tilting finiteness is preserved under taking quotients.

Recall that an algebra $\Lambda$ is minimal representation-infinite (or min-rep-infinite, for short) if $\Lambda$ is rep-infinite and any proper quotient of $\Lambda$ is representation-finite. Following our notations in [Mo2], by $\operatorname*{Mri}({\mathfrak{F}_{\operatorname*{sB}}})$ we respectively denote the family of min-rep-infinite special biserial algebras and $\operatorname*{Mri}({\mathfrak{F}_{\operatorname*{nD}}})$ denotes the family of non-distributive min-rep-infinite algebras. Before we summarize the relevant results on the brick (in)finiteness of these algebras, let us recall that in [Mo1], the following bound quivers are called generalized barbell:

where $I=\langle\beta\alpha,\delta\gamma\rangle$, $C_{L}=\alpha\cdots\beta$ and $C_{R}=\gamma\cdots\delta$ are cyclic strings with no common vertex, except for possibly the case where $\mathfrak{b}$ is of zero length (which implies $x=y$). Moreover, $\mathfrak{b}$ (respectively $C_{L}$ and $C_{R}$) can have any length (respectively any positive length) and arbitrary orientation of their arrows, provided $C_{R}C_{L}$ is not a uniserial string in $(Q,I)$. We note that generalized barbell quivers are a slight generalization of “barbell” quivers introduced by Ringel [Ri1], where he always assume the bar $\mathfrak{b}$ is of positive length.

The following theorem summarizes some of our earlier results on the study of $\tau$-tilting finiteness. To make them more congruent with the scope of this paper, below we state them in terms of bricks.

We remark that $\operatorname*{Mri}({\mathfrak{F}_{\operatorname*{sB}}})$ and $\operatorname*{Mri}({\mathfrak{F}_{\operatorname*{nD}}})$ consist of only tame algebras and either of these two families contains both brick-finite and brick-infinite algebras (see [Mo1, Mo2] for full classifications).

We also recall that an algebra $\Lambda$ is said to be minimal $\tau$-tilting infinite if $\Lambda$ is $\tau$-tilting infinite but every proper quotient of $\Lambda$ is $\tau$-tilting finite. From Theorem 2.2, it is immediate that minimal $\tau$-tilting infinite algebras are the same as minimal brick-infinite algebras. Here we only list some of the main properties of these algebras and for more details we refer to [MP]. Recall that $\Lambda$ is called central provided its center is the ground field $k$.

To highlight some fundamental differences between these modern and classical notions of minimality, we remark that min-rep-infinite algebras are not necessarily central and their bound quivers can have several nodes. Furthermore, note that although $\tau$-tilting finiteness is preserved under algebraic quotients, there exists tilting-finite algebra $\Lambda$ such that $\Lambda/J$ is tilting-infinite, for an ideal $J$ in $\Lambda$.

In this subsection we collect some basic tools used in this paper which allow us to move between the algebraic and geometric sides of our problem. In particular, for algebra $\Lambda$ and a dimension vector $\underline{d}$ in $\mathbb{Z}^{|\Lambda|}_{\geq 0}$, let $\operatorname*{rep}(\Lambda,\underline{d})$ denote the affine (not necessarily irreducible) variety parametrizing the modules in $\operatorname*{mod}(\Lambda,\underline{d})$. Here, $\operatorname*{mod}(\Lambda,\underline{d})$ denotes the subcategory of $\operatorname*{mod}\Lambda$ consisting of all modules of dimension vector $\underline{d}$.

Under the action of $\operatorname*{GL}(\underline{d})$ via conjugation, $\operatorname*{rep}(\Lambda,\underline{d})$ can be viewed as a scheme, as well as an affine variety, where the orbits of this action are in bijection with the isomorphism classes of modules in $\operatorname*{mod}(\Lambda,\underline{d})$. Through this conceptual dictionary, we study some geometric properties of representations of the bound quiver $(Q,I)$, where $\Lambda=kQ/I$ is an admissible presentation of $\Lambda$. For $M$ in $\operatorname*{mod}(\Lambda,\underline{d})$, by $\mathcal{O}_{M}$ we denote the $\operatorname*{GL}(\underline{d})$-orbit of $M$, when it is viewed as a point in $\operatorname*{rep}(\Lambda,\underline{d})$. If $\operatorname*{rep}(\Lambda,\underline{d})$ is viewed as the $k$-points of a corresponding scheme, it is known that $\mathcal{O}_{M}$ is open in this scheme if and only $M$ is rigid. However, if we consider $\operatorname*{rep}(\Lambda,\underline{d})$ as a variety, there could be non-rigid modules $N$ such that $\mathcal{O}_{N}$ is open. Although both of these geometric structures are rich and come with powerful tools, we mostly treat $\operatorname*{rep}(\Lambda,\underline{d})$ as an affine variety. When there is no risk of confusion, $\operatorname*{mod}(\Lambda,\underline{d})$ is referred to as a variety to reflect the geometric structure that comes from $\operatorname*{rep}(\Lambda,\underline{d})$.

Let $\operatorname*{ind}(\Lambda,\underline{d})$ and $\operatorname*{brick}(\Lambda,\underline{d})$ respectively denote the set of all indecomposable modules and bricks in $\operatorname*{mod}(\Lambda,\underline{d})$. It is known that $\operatorname*{brick}(\Lambda,\underline{d})$ is an open subset of $\operatorname*{mod}(\Lambda,\underline{d})$. Let $\operatorname*{Irr}(\Lambda,\underline{d})$ be the set of all irreducible components of $\operatorname*{mod}(\Lambda,\underline{d})$, and by $\operatorname*{Irr}(\Lambda)$ we denote the union of all $\operatorname*{Irr}(\Lambda,\underline{d})$, where $\underline{d}$ is an arbitrary dimension vector. A component $\mathcal{Z}\in\operatorname*{Irr}(\Lambda)$ is called indecomposable provided it contains a non-empty open subset $U$ which consists of indecomposable representations. In [CBS], the authors prove a geometric analogue of the Krull-Schmidt decomposition for irreducible components, which highlights the role of indecomposable components among all irreducible ones.

For each $\mathcal{Z}$ in $\operatorname*{Irr}(\Lambda)$, the algebraic properties of the modules in $\mathcal{Z}$ capture important information on the geometry of $\mathcal{Z}$, and vice versa. Motivated by this interaction, Chindris, Kinser and Weyman [CKW] have recently adopted a geometric approach to generalize the notion of representation-finiteness, primarily based on the properties of irreducible components. In particular, $\Lambda$ is said to have dense orbit property provided every $\mathcal{Z}$ in $\operatorname*{Irr}(\Lambda)$ contains a dense orbit. By some simple geometric considerations, one can show that every rep-finite algebra has the dense orbit property. In [CKW], the authors show that the new notion is novel and construct explicit rep-infinite algebras which have the dense orbit property. Furthermore, they prove that a string algebra (and more generally, each special biserial algebra) is rep-finite if and only if it has the dense orbit property.

Adopting this algebro-geometric approach, we say that $\Lambda$ is brick-discrete if for each $d\in\mathbb{Z}_{\geq 0}$, there are only finitely many (isomorphism classes of) bricks of dimension $d$. This is equivalent to the fact that for each $\mathcal{Z}$ in $\operatorname*{Irr}(\Lambda)$, if $M$ belongs to $\operatorname*{brick}(\mathcal{Z})$, then $\mathcal{Z}=\overline{\mathcal{O}}_{M}$. Here, $\overline{\mathcal{O}}_{M}$ denotes the orbit closure of $\mathcal{O}_{M}$. We remark that brick-discrete algebras have been treated in [CKW], where the authors introduced them under the name “Schur representation-finite” algebras. To avoid confusion between brick-finite and Schur representation-finite algebras, we use our new terminology and call the latter type brick-discrete.

In [CKW], it is shown that if $\Lambda$ has the dense orbit property, then it is brick-discrete, but the converse does not hold in general. So, brick-discreteness was considered as a generalization of the dense orbit property, and hence a generalization of rep-finiteness. We observe that every brick-finite algebra is brick-discrete. In contrast, in general it is not known whether brick-discrete algebras are necessarily brick-finite. In fact, this is the content of the following conjecture, which is a precursor of Conjecture 1.1.

The above conjecture first appeared in the arXiv version of [Mo2], where the first-named author proposed an algebro-geoemtric realization of $\tau$-tilting finiteness. Moreover, it is verified for all algebras treated in that paper. We also remark that the numerical implication of the above conjecture was later stated in [STV].

For a special biserial algebra $\Lambda$, an irreducible component $\mathcal{Z}$ in $\operatorname*{Irr}(\Lambda)$ is called a string component if it contains a string module $M$ such that $\mathcal{O}_{M}$ is dense in $\mathcal{Z}$. That being the case, we get $\mathcal{Z}=\overline{\mathcal{O}}_{M}$, which implies that $\mathcal{Z}$ can be specified by the isomorphism class of the string module $M$. In contrast, $\mathcal{Z}$ is a band component provided $\mathcal{Z}$ contains a family $\{M_{\lambda}\}_{\lambda\in k^{*}}$ of band modules such that $\bigcup_{\lambda\in\mathcal{C}}\mathcal{O}_{M_{\lambda}}$ is dense in $\mathcal{Z}$. In this case, $\mathcal{Z}=\overline{\bigcup_{\lambda\in\mathcal{C}}\mathcal{O}_{M_{\lambda}}}$. Hence, a band component is determined by the band that gives rise to the one-parameter family $\{M_{\lambda}\}_{\lambda\in k^{*}}$. Provided $\Lambda$ is a string algebra, $\operatorname*{Irr}(\Lambda)$ consists only of string and band components.